l13 - the state equation in the time domain we can take the...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
16.06 Lecture 13 More State Space Modeling and Transfer Function Matrices John Deyst October 2, 2003 Today’s Topics 1. More state space modeling. 2. Laplace Transforms for vector/matrix differential equations. 3. An example 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
A special case for state space models is when there are as many zeros as poles. For example if we have Recall that we first break G ( s ) into two parts The first transfer function relates the input r to the intermediate output x 1 . In the time domain this implies the D.E. 2
Background image of page 2
So the state D.E.’s are Hence Implying the following block diagrams 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
However now the output is which, in the time domain is If we go back to the block diagram we see that we must feed the ˙ x 2 signal forword to get the output We can then write the equation for w 4
Background image of page 4
Laplace Transforms of State Vectors With appropraite definitions of terms we can usefully apply Laplace Transform methods to vector/matrix differential equations. Thus we define the Laplace Transform of a state vector as It then follows naturally that the L.T. of the derivative of x ( t )is 5
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Thus if we have
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: the state equation in the time domain we can take the L.T. of both sides to obtain which can also be written as where I is the identity matrix 6 We can then solve for the L.T. of the state as Now we also have the output equation which can also be transformed to obtain the L.T. of the output vector Upon substitution of the equation for the L.T. of the state obtains 7 If we are only interested in the response to inputs we can obtain a matrix transfer function where Example Earlier we turned the scalar transfer function into a state space model and found the following matrices 8 Then Its inverse is Note that the denominater is the left hand side of the characteristic equation Also 9...
View Full Document

Page1 / 9

l13 - the state equation in the time domain we can take the...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online